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Properties

Label 2-459-153.106-c1-0-9
Degree $2$
Conductor $459$
Sign $0.451 - 0.892i$
Analytic cond. $3.66513$
Root an. cond. $1.91445$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.92 + 1.10i)2-s + (1.45 + 2.52i)4-s + (0.680 + 0.182i)5-s + (2.07 − 0.556i)7-s + 2.04i·8-s + (1.10 + 1.10i)10-s + (−1.34 + 0.360i)11-s + (0.789 + 1.36i)13-s + (4.60 + 1.23i)14-s + (0.657 − 1.13i)16-s + (−0.767 + 4.05i)17-s − 0.618i·19-s + (0.532 + 1.98i)20-s + (−2.98 − 0.798i)22-s + (−0.903 + 3.37i)23-s + ⋯
L(s)  = 1  + (1.35 + 0.784i)2-s + (0.729 + 1.26i)4-s + (0.304 + 0.0815i)5-s + (0.784 − 0.210i)7-s + 0.721i·8-s + (0.349 + 0.349i)10-s + (−0.405 + 0.108i)11-s + (0.218 + 0.379i)13-s + (1.23 + 0.329i)14-s + (0.164 − 0.284i)16-s + (−0.186 + 0.982i)17-s − 0.141i·19-s + (0.119 + 0.444i)20-s + (−0.635 − 0.170i)22-s + (−0.188 + 0.703i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(3.66513\)
Root analytic conductor: \(1.91445\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1/2),\ 0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61403 + 1.60785i\)
\(L(\frac12)\) \(\approx\) \(2.61403 + 1.60785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (0.767 - 4.05i)T \)
good2 \( 1 + (-1.92 - 1.10i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.680 - 0.182i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.07 + 0.556i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.34 - 0.360i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.789 - 1.36i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 0.618iT - 19T^{2} \)
23 \( 1 + (0.903 - 3.37i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.71 + 10.1i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (6.01 + 1.61i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 + (-0.573 + 2.14i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.30 - 2.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.53 + 7.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (-5.64 + 3.25i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.85 - 0.765i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.68 + 9.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.47 + 5.47i)T - 71iT^{2} \)
73 \( 1 + (-4.12 + 4.12i)T - 73iT^{2} \)
79 \( 1 + (7.43 - 1.99i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.68 + 1.55i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.02T + 89T^{2} \)
97 \( 1 + (4.51 + 16.8i)T + (-84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47401743215686876018130758272, −10.49977837013861379326900212522, −9.385723478968790951158652065925, −8.045004932605389128719014156310, −7.42965489996910369379145656128, −6.22912310332799484723772855290, −5.61383821549695642246494884355, −4.50631421608278635994374135295, −3.73995211176008239367703585853, −2.08699429884232938178016636614, 1.71582785641283146746764518306, 2.85744824333333144559321225305, 4.01240951659053631229838383996, 5.19811452462364304998447018547, 5.56355666360427030368731860429, 7.00117605332359937190421560101, 8.202219726853843043967538393722, 9.244764318747041127456703220128, 10.49949464985746467935989886814, 11.08366611472603341204853046084

Graph of the $Z$-function along the critical line